# The Theory of Risk and Risk Aversion

## The Theory of Risk and Risk Aversion

CHAPTER 3 The Theory of Risk and Risk Aversion Jack Meyer Department of Economics, Michigan State University, USA Contents 3.1  Introduction 99 3...

CHAPTER

3

The Theory of Risk and Risk Aversion Jack Meyer

Department of Economics, Michigan State University, USA

Contents 3.1  Introduction 99 3.2  The Mean-Variance Approach to Risk and Risk Aversion 100 3.3  The Expected Utility Approach 104 3.4  Risk Aversion 106 3.5  Increases in Risk and Stochastic Dominance 122 3.6  Mean Variance Analysis and Expected Utility 129 References 131

Abstract Risk and risk aversion are important concepts when modeling how to choose from or rank a set of random variables. This chapter reviews and summarizes the definitions and related findings concerning risk aversion and risk in both a mean-variance and an expected utility decision model context.

Keywords Risk Aversion, Increases in Risk, Stochastic Dominance

JEL Classification Codes D81

3.1 INTRODUCTION Risk and risk aversion have been recognized and included in the scientific discussion of decision making under uncertainty for hundreds of years. The St. Petersburg Paradox presented by mathematician Daniel Bernoulli (1954) in 1738 is often mentioned when first introducing the topic. Bernoulli identifies a gamble that most persons would pay only a small sum to be allowed to participate in, but whose expected value is infinite. Bernoulli suggests diminishing marginal utility as a reason why this is the case. Bernoulli describes the following gamble. Suppose a person receives an outcome of size 2 if flipping a fair coin results in heads on the first toss. The gamble ends if this occurs. Should a tail occur on the first toss, the gamble continues. When the coin is flipped a second time, if heads is observed, the payment is increased to 4, and the gamble ends. Should a tail occur on each of the first two tosses, the gamble continues Handbook of the Economics of Risk and Uncertainity, Volume 1 ISSN 2211-7547, http://dx.doi.org/10.1016/B978-0-444-53685-3.00003-9

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to a third toss, with the payoff and the continuation decision following this same pattern. The gamble continues until heads is first observed. The payment is 2n when heads is first observed on the nth toss. The probability of heads occurring first on the nth toss i 1 of a fair coin is {1/2}n. Thus, the expected value of this gamble is given by ∞ i=l (2 )( 2i ) which is not finite. The paradox that Bernoulli is pointing out is that most persons are unwilling to pay more than a modest sum to participate in this gamble even though the expected value from participating in the gamble is infinite. Clearly, then, using only the expected value to rank gambles is not enough. Bernoulli uses the St. Petersburg Paradox to suggest that the gain in utility from additions to wealth diminishes as a person is wealthier. In fact he suggests that the utility function could be represented by the logarithmic function. Since this function is concave, this solution to the St. Petersburg Paradox has been pointed to as the first representation of risk aversion in an expected utility decision model. The focus in this chapter is on how risk aversion and risk are represented in various decision models. The emphasis is overwhelmingly on the expected utility decision model discussed in Sections 3.3 to 3.5. The mean-variance decision model is also discussed, first in Section 3.2 and then again briefly at the end in Section 3.6.

3.2 THE MEAN-VARIANCE APPROACH TO RISK AND RISK AVERSION 3.2.1 Introduction One of the primary alternatives to the expected utility decision model in applied economic analysis is the mean-variance decision model (M-V). In this model the risk associated with a random variable is measured by the variance.1 The typical notation is to use μ to represent the mean or expected outcome from a random variable, and to use σ or σ2 to represent the standard deviation, or variance, respectively. Any random variable for which σ is strictly greater than 0 is said to be risky. Only a degenerate random variable, one where μ occurs with certainty, is without risk. Variance and aversion to variance have been used to represent risk and risk aversion for many years and continue to be used extensively especially in disciplines other than economics. Researchers in statistics regularly determine the minimum variance unbiased estimator of a parameter. Those in finance calculate and use the covariance of an asset’s return with the market portfolio as a way to measure the portion of an asset’s risk that cannot be eliminated by diversification. The price of risk is often given as the gain in expected return from assuming an additional unit of standard deviation. While economics uses the M-V decision model less often than it did before the expected utility decision model was fully developed, the M-V model is still employed in some areas and 1

 ore often, the positive square root of this variance, the standard deviation, is used so that the unit of measure for risk M and the unit of measure for the mean value are the same.

The Theory of Risk and Risk Aversion

for certain questions. The version of the M-V decision model described in the following sections draws heavily on the writings of Markowitz (1952, 1959) and Tobin (1958).

3.2.2  The M-V Decision Model The critical and powerful assumption that is made in the M-V decision model is that preferences over random variables can be represented by a utility or ranking function denoted V(σ, μ) that depends only on the mean and standard deviation of the possible outcomes. This ranking function represents an ordinal ranking of random variables, and thus the function V(σ, μ) is unique up to a positive monotonic transformation. The ordering over random variables represented by V(σ, μ) is complete; that is, all pairs of random variables can be ranked as one better than the other, or the decision maker is indifferent between the two random variables. The focus in this M-V decision model is on just two moments of the probability distributions. All other moments and other properties of these probability distributions such as the median, the skewness, the kurtosis, or the smallest possible outcome play no separate role in the ranking of random variables. It is usual in economic analysis to model decisions so that higher values for the outcome variable are preferred to lower values. This is clearly the case for decisions where the outcome is wealth, or profit, or consumption, or return on an investment. This preference for higher values is reflected in the assumption that Vμ > 0 for all values of σ and μ. Risk aversion is exhibited when lower standard deviation or variance is preferred to higher standard deviation or variance; that is, when Vσ < 0 for all σ and μ. Risk loving behavior is represented when Vσ > 0 for all σ and μ, and risk neutrality, not caring about risk, is characterized by Vσ = 0 for all σ and μ. Each of these three types of risk preference places a requirement on the partial derivative Vσ for all σ and μ.Thus, the definitions leave open the possibility that preferences can display risk aversion for some σ and μ, and be risk loving for other values; that is the M-V decision model allows for decision makers who are neither risk averse nor risk loving nor risk neutral, but instead display each of these preferences toward risk at different values for σ and μ. A frequently used form for an M-V utility function is V(σ, μ) = μ − λ·σ2 although there are others that are employed. For this particular form, λ > 0 characterizes the decision maker as risk averse. This form and others that are used are selected to ensure that V(σ, μ) is a quasiconcave function. This implies that maximization of V(σ, μ) subject to choosing from a linear opportunity set in (σ, μ) space typically yields an interior solution. It is most common to place σ on the horizontal axis and μ on the vertical axis when graphing indifference curves for these utility functions. Figure 3.1 illustrates typical indifference curves in (σ, μ) space labeled as I1 and I2. One of the positive features of the M-V decision model is that it is very simple and lends itself to two-dimensional graphical analysis and illustration. Opportunity or choice sets are sets of points in two-space, with (σ, μ) denoting the coordinates. For

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µ

I2

I1

ρ

0

σ

Figure 3.1  Typical indifference curves in (σ, μ) space, labeled as I1 and I2.

example, in the one risky and one riskless asset portfolio model with unit initial wealth, and where ρ is the riskless return and σr and μr are the risk and expected return for the risky asset, the opportunity set in (σ, μ) space is a straight line with vertical intercept (0, ρ) and slope equal to µrσ−ρ.This is also illustrated in Figure 3.1.The value of this slope r is referred to as the price of risk in the M-V decision model.This price of risk represents the extra expected return over the risk-free return that is earned by assuming an additional unit of risk as measured by the standard deviation of return. The opportunity set in this portfolio decision is a weakly convex set, and for most economic decision models, assumptions are made so the opportunity set in (σ, μ) space is a convex set. Because of its simplicity and because of its ability to take covariances of assets into account, the M-V decision model is often one of the first models used when beginning the discussion of a particular decision under uncertainty. The M-V model is particularly well suited for and difficult to replace when modeling decisions where forming portfolios of assets or activities is possible, and where the fundamental decision being examined is which portfolio would be chosen by the decision maker. This point was emphasized by Markowitz (1959) who titled his book Portfolio Selection. The ability to deal with portfolio questions continues to be an important positive property of the M-V decision model. Because of the role that portfolios play in that model, the M-V decision model is at the heart of the well-known Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner (1965) and Mossin (1966).

3.2.3  M-V Model Weaknesses The M-V decision model has been replaced by the expected utility (EU) decision model in theoretical economic analysis of most decision problems, especially those where the selection of an optimal portfolio is not one of the decisions of interest. Fundamental

The Theory of Risk and Risk Aversion

weaknesses of the M-V model lead many to use the EU model instead. One weakness is that in the M-V decision model, all alternatives with the same mean and variance are necessarily ranked the same. This implies, for instance, that alternatives with positively skewed outcomes are ranked the same as ones which are negatively skewed as long as the mean and variance of the two are the same. For example, alternative A: obtaining −1 with probability .9999 and 9999 with probability .0001 is ranked the same as alternative B: obtaining −9999 with probability .0001 and 1 with probability .9999. These two alternatives each have the same mean value, 0, and the same variance, 9999, and therefore all M-V decision makers must be indifferent between A and B when preferences are represented by V(σ, μ). There is much evidence indicating that these two alternatives are not ranked the same by many decision makers, and that decision makers vary in their preference for such skewness. This same example also illustrates the fact that the size of the worst possible outcome is not important in the M-V ranking of alternatives. For many decision makers, the possibility of losing 9999 in gamble B makes it much worse than gamble A where the maximum loss is only 1 and this preference cannot be represented using an M-V utility function V(σ, μ). Another important negative property of the M-V decision model was pointed out by Borch (1969), who shows that indifference curves in (σ, μ) space are not consistent with preferring higher outcomes. The following example illustrates a general calculation presented by Borch. Suppose a decision maker is indifferent between two points, (σA, μA) = (1, 1) and (σB, μB) = (2, 2). Two gambles A and B can be constructed with these values for σ and μ such that indifference between the two gambles is not possible when decision makers prefer higher outcomes. For this particular choice of (σA, μA) and (σB, μB), the constructed gamble A has outcomes 0 or 2 which are equally likely, while gamble B yields either 0 and 4 with equal probability. These two gambles satisfy (σA, μA) = (1, 1) and (σB, μB) = (2, 2), yet any person who prefers larger outcomes to smaller ones must strictly prefer gamble B to gamble A. It is not possible that (1, 1) and (2, 2) lie on the same indifference curve if these two random variables are to be ranked. Borch gives a general procedure that allows such an example to be constructed for any two points claimed to be on an indifference curve in (σ, μ) space. Using terminology not available at the time Borch was writing, Borch shows that for any two points on an indifference curve for utility function V(σ, μ), two random variables can be constructed with the appropriate values for σ and μ, where one dominates the other in the first degree. First degree stochastic dominance is discussed in Section 3.5.2. The M-V decision model exhibits the negative properties just mentioned. The EU decision model on the other hand is more flexible and can represent decision makers who exhibit preference for, or aversion to, skewness, or those who focus intensely on the worst outcomes, or those who always prefer larger outcomes. In general, by choosing the form for the utility function appropriately, EU risk preferences can be made to reflect preference for, or avoidance of, virtually any property or attribute of a probability

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distribution or random variable that may be important to the decision maker. In the last section of this chapter, Section 3.6, the M-V model is again discussed very briefly. The focus in that discussion is on the question of when do the M-V and EU decision models coincide. The EU model is discussed next.

3.3 THE EXPECTED UTILITY APPROACH Determining the ranking function for random variables in the expected utility (EU) decision model is a two-step process. First, a utility function u(x), whose domain is the set of possible outcomes that the random variables can yield, is determined. Various properties of this utility function represent the risk preferences of the specific decision maker. The second step in the expected utility ranking procedure calculates the expectation of this utility function u(x). This expectation, expected utility, is the function used to represent the ranking of random variables. Common notation for continuously distributed random variables is to let F(x) denote the cumulative distribution function (CDF) associated with a particular random variable. The expected utility or ranking number attached to that random variable is given by:  b u(x)dF(x) U(F(x)) = EF u(x) = a

where the support of the random variable is assumed to lie in the interval [a, b]. Alternatively, for discrete random variables f(xi) is used to denote the probability function indicating the likelihood of outcome xi, where i = 1, 2, …, n. The expected utility is then given by n U(f(xi )) = Eu(x) = u(xi ) · f(xi ). i=1

The magnitude of this ranking number, U(F(x)) or U(f(xi)), is then compared with a similar number for another CDF G(x) or another probability function g(xi)  b U(G(x)) = EG u(x) = u(x)dG(x) an U(g(xi )) = Eu(x) = u(xi ) · g(xi ) i=1

to choose between or rank the two random variables represented by CDFs F(x) and G(x) or probability functions f(xi) and g(xi). As with the M-V ranking of random variables, this ranking provided by EU is an ordinal one; thus U(F) or U(f) is unique to a positive monotonic transformation. The von-Neumann-Morgenstern (N-M) utility function u(x), however, is unique to a

The Theory of Risk and Risk Aversion

positive linear transformation. This property results from the fact that the computing of expected utility is a process that is linear in probabilities. The fact that u(x) is unique to a positive linear transformation does not imply that u(x) represents a cardinal ranking over outcomes x, nor that U(F) represents a cardinal ranking. Early in the discussion of the expected utility decision model there was confusion concerning these ordinality/cardinality issues. Alchian (1953), Baumol (1958) and Luce and Raiffa (1957) each discuss this point. A number of papers present sets of axioms or assumptions concerning preferences for random variables that are necessary and sufficient for the existence of a N-M utility function u(x) with which expected utility is computed. Herstein and Milnor (1953), Friedman and Savage (1948) and Arrow (1965) are examples. Other chapters in this handbook discuss the EU assumptions more fully so only an informal presentation is given here. Luce and Raiffa provide a particularly easy-to-follow proof of the basic representation theorem, and a brief presentation of the logic of their argument is given next. The Luce and Raiffa proof is proof by construction. Their analysis demonstrates the existence of the N-M utility function u(x) and how one uses it to rank random variables by giving a step-by-step procedure for constructing u(x). Suppose outcomes x1, x2, … xn are possible. These outcomes can be elements of real number space, vectors representing bundles of goods, or even unrelated objects such as tickets to a Detroit Tigers baseball game or a cheese enchilada dinner at Jose’s restaurant. To determine a N-M utility function for calculating EU, several assumptions are made. First, it is assumed that these outcomes can be ordered so that x1 is the most preferred, xn is the least preferred, and xi is preferred or indifferent to xj for i ≤ j. Once this ordering of outcomes has occurred, the next assumption is that for each outcome xi one can find a probability ui such that a particular gamble x˜ i which is indifferent to xi can be formulated. This gamble x˜ i has only two possible outcomes, x1 and xn, and yields the most preferred outcome x1 with probability ui and the least preferred outcome xn with probability (1 − ui). Obviously ui = 1 for outcome x1, and ui = 0 for outcome xn, and ui is between zero and one for all other outcomes. This assumption is a continuity assumption. Having constructed a gamble x˜ i that is indifferent to xi, the important substitution assumption is that it is possible to replace xi with x˜ i wherever xi occurs as a possible outcome in a gamble. Doing this for all xi and assuming the usual rules of probability allows any gamble to be reduced to one that has only x1 and xn as possible outcomes. For example, let (p1, p2, …pn) and (q1, q2, …qn) represent two gambles over outcomes x1, x2, … xn. In each of these gambles, replace outcomes x2 to xn−1 by x˜ 2 to x˜ n−1. Doing this yields two equivalent gambles with only x1 and xn as possibilities. Moreover,  the probability of receiving the most preferred outcome x1 is ni=1 ui · pi for the first  gamble and ni=1 ui · qi for the second. With only two possible outcomes, the gamble with the highest probability of obtaining the most preferred outcome is best. Thus, the

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term ni=1 ui · pi can be used to represent the ranking of gambles. This of course is the expected utility for that gamble when function u(xi) = ui is chosen to be the N-M utility function. The expected utility decision model, although available in its modern axiomatic form since the work of von Neumann and Morgenstern (1944), did not dominate and replace the M-V decision model in economic analysis until the 1960s.2 Important papers and books by Arrow (1965), Pratt (1964), Rothschild and Stiglitz (1970), Hadar and Russell (1969), Hanoch and Levy (1969) and many others led to this transition. Arrow and Pratt present definitions and measures of risk aversion, while Rothschild and Stiglitz define increases in risk and mean preserving spreads to replace the use of variance as a measure of risk. Hadar and Russell and Hanoch and Levy define first and second degree stochastic dominance. The focus of the discussion of the EU decision model in this chapter is on defining and measuring risk aversion and risk, and begins with a definition of risk aversion, and the well-known Arrow-Pratt (A-P) measure of absolute risk aversion.

3.4 RISK AVERSION 3.4.1 Introduction A decision maker is said to be risk averse if that person starting from a position of certainty rejects the addition of any fair gamble to that certain starting position.3 Adding a fair gamble to a nonrandom starting position yields a gamble whose mean value is the same as the initial nonrandom starting value. Thus, a certain starting position is converted to a random one with the same mean. Risk aversion is always avoiding such a change. This simple and basic definition is used to characterize risk aversion in virtually every decision model that ranks random variables. In the M-V model just discussed, the assumption that Vσ < 0 for all σ and μ ensures that all fair gambles are rejected and that the decision maker is risk averse. In the EU decision model, risk aversion is equivalent to the concavity of the N-M utility function u(x) used to compute expected utility. Consider any gamble z˜ with outcomes x and y, where x and y occur with probabilities p and 1 − p, respectively. The mean value of this gamble is μ = p · x + (1 − p) · y, and the utility from µ is given by u(μ). The expected utility from the gamble is [p · u(x) + (1 − p)u(y)] = Eu(˜z). For functions u(z) that are concave, u(p · x + (1 − p) · y)  ≥ [p · u(x) + (1 − p)u(y)]. This property of concave functions is illustrated in Figure 3.2 and is a form of Jensen’s inequality which indicates directly that Eu(˜z) ≤ u(µ) for all concave functions u(z). All risk averse persons prefer to receive the mean value of a gamble, rather than participate in the gamble itself. If the utility function were convex rather than concave, the argument just given and the use of Jensen’s inequality is 2 There 3 A

were earlier important papers including a paper by Friedman and Savage (1948). fair gamble is any gamble whose mean value is zero.

The Theory of Risk and Risk Aversion

u(y) u(µ) Eu u(x)

0

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y

µ

Figure 3.2  Illustration that Eu(˜z) ≤ u(µ) for all concave functions u(z).

reversed. For the linear or risk neutral utility function, Eu(˜z) = u(µ) for all random variables. Since u(z) is unique to a positive linear transformation, the most often used utility function to represent risk neutrality is the identity function u(z) = z. Risk neutral decision makers choose among random variables on the basis of E[˜z] = µ, their mean values. Thus, the M-V and EU decision models are the same in this regard. Also like the M-V model, the EU decision model allows for decision makers who are neither risk averse, nor risk loving, nor risk neutral; that is the utility function u(z) can be neither concave, convex, nor linear for all z.

3.4.2  Measuring Risk Aversion Locally Arrow and Pratt (A-P) not only identify risk aversion with concavity of the N-M utility function u(x), they also provide a way to measure the degree of concavity, and hence the strength or intensity of risk aversion. A-P give two related measures and both are extensively used in current day economic analysis. Most of the material discussed here focuses on the A-P measure of absolute risk aversion, denoted Au(x). The relative risk aversion measure, Ru(x), is briefly discussed after Au(x) is carefully described.

Definition 3.1:  The absolute risk aversion measure Au(x) for N-M utility function u(x) is ′′

Au (x) =

−u (x)

. ′ u (x) Two things concerning Au(x) are worth noting at the outset. First, absolute risk aversion is defined for outcomes in single dimension real number space. The risk aversion measure is a univariate function. Arrow and Pratt refer to the outcome variable x as

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wealth. Thus, even though the expected utility ranking of random variables allows outcomes to be vectors or dissimilar objects, this univariate measure of risk aversion does not. Second, Au(x) is uniquely associated with a given set of risk preferences; that is, u(x), or any positive linear transformation of u(x), each lead to exactly the same measure of absolute risk aversion. The process of differentiation and forming a ratio of derivatives eliminates the arbitrary constants associated with N-M utility functions.Thus, no matter which of the functions A + B · u(x) for B > 0 is used to represent the risk preferences of a decision maker, the A-P absolute risk aversion measure for that decision maker is a single function independent of such affine transformations. In a variety of settings involving small risks, both Arrow and Pratt show that a decision maker’s reaction to a risk is proportional to the product of this absolute risk aversion measure Au(x) and the size of the risk. For instance, the risk premium, π(x), is defined as the amount a decision maker starting with nonrandom wealth x would discount the mean value of a random variable in order to sell that random variable. Letting z˜ represent a random variable with mean μz, the risk premium π(x) when starting with nonrandom wealth x for random variable z˜ is defined by: u(x + µz − π(x)) = Eu(x + z˜ ).

It is common to restrict attention to variables z˜ whose mean value is zero which allows this expression to be written as: u(x − πz (x)) = Eu(x + z˜ ).

This definition for the risk premium π(x) holds for all z˜ no matter the size of the possible outcomes that can occur for gamble z˜. For small z˜, however, Arrow and Pratt use Taylor Series approximations to show that π(x) is approximated by 2

π(x) ≈ (1/2) · Au (x) · σz .

Putting this mathematical statement into words, the decision maker’s reaction to risk, π(x), is proportional to the product of the measure of risk aversion Au(x), and the measure of risk σz2. For random variables whose outcomes are negative, the insurance premium, i(x), can be considered instead. Here, i(x) represents the amount above the expected loss for a random variable that the decision maker would pay to be fully insured against that risk. When μz = 0, i(x) and π(x) are the same, and in general, i(x) = π(x) + μz. A-P also present a related finding concerning a quite different reaction to risk called the probability premium, denoted φ(x, h). The probability premium is defined to be the deviation from 1/2 that is needed to induce a decision maker to accept a gamble with two possible outcomes that are equal in size, but opposite in sign. The notation used is that an amount h is either added to or subtracted from current wealth which is nonrandom value x. Formally, the equation that defines the probability premium φ(x, h) is:

The Theory of Risk and Risk Aversion

u(x) = [.5 + φ(x, h)]u(x + h) + [.5 − φ(x, h)]u(x − h).

Again this definition for φ(x, h) holds for all values for h. For small h, Arrow and Pratt use a quadratic Taylor Series approximation to u(x + h) and u(x − h) and show that the probability premium is approximately given by: φ(x, h) ≈ (1/2) · Au (x) · h.

As with the risk premium, the form of this result is that the decision maker’s reaction to risk, φ(x, h), is proportional to the product of the strength or intensity of risk aversion Au(x), and the size of the risk, in this case measured by h.

3.4.3  A-P Global Risk Aversion The local and approximate findings discussed in Section 3.4.1 lend considerable intuitive appeal to Au(x) as a measure of the intensity of risk aversion. These findings are greatly enhanced, however, by global or in the large results that do not rely on the assumptions that the gambles being considered are small.These global results, stated next, are the primary justification for using Au(x) as an intensity measure for risk aversion in the EU decision model. The global results take on a different form than do the local results, and involve the comparison of the risk aversion levels of two decision makers with utility functions u(x) and v(x). The theorem given below defines u(x) as at least as risk averse as v(x) for all x using the condition Au(x) ≥ Av(x) for all x. The theorem and the discussion that follows also provide several other ways to characterize the A-P more risk averse partial order over decision makers.

Theorem 3.1:  The followiwng statements are equivalent. (1) u(x) is at least as risk averse as v(x). (2) Au(x) ≥ Av(x) for all (x). (3) u(x) = θ(v(x)) where θ(·) ≥ 0 and θ″(·) ≤ 0. (4) πu(x) ≥ πv(x) for all x and all gambles z˜ . (5) φu(x, h) ≥ φv(x, h) for all x and all h. The first two of these statements give Arrow and Pratt’s definition of more risk averse. The third is a mathematical characterization indicating that u(x) is more risk averse than v(x) if and only if u(x) can be written as an increasing and concave transformation of v(x). An alternate mathematical characterization that is sometimes useful involves marginal utility functions rather than utility functions and is: ′

(6)  u (x) = δ (x) · v (x) where δ (x) ≥ 0 and δ″(x) ≤ 0. The characterization given in (6) implies that Au(x) = Av(x) + Aδ(x), where ′′ Aδ (x) = −δ′ (x) ≥ 0. δ (x)

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Characterizations (4) and (5) extend the local results concerning the risk and probability premiums to risks which are no longer assumed to be small. A more risk averse person always has a larger risk and probability premium. A rewriting of characterization (4) allows the condition on risk premiums to be stated for certainty equivalents instead. The certainty equivalent (CE) for a gamble is defined to be that amount which, when obtained for certain, provides the same expected utility as the gamble. Letting CEu and CEv denote the certainty equivalents for persons with utility functions u(x) and v(x) respectively, (7)   u(x + CEu ) = Eu(x + z˜ ) and v(x + CEv ) = Ev(x + z˜ ) imply CEu ≤ CEv for all x and all z˜. It is the case that the reactions to risk in the form of the risk premium or probability premium each involve a variable which is constructed to allow measurement of the intensity of risk aversion. It is also possible to consider other reactions to risk that involve choices made by the decision maker in response to risk. In many decision settings, the choice of a parameter is also a way to define or characterize the Pratt and Arrow absolute risk aversion measure. Pratt and Arrow provide this discussion for the investment choice made in the single risky and riskless asset portfolio model.4 The literature contains many similar findings in other decision contexts. Consider a decision maker who can allocate current nonrandom wealth between a risky  = α · ˜r + (W0 − α) · ρ, where ˜r is and riskless asset. Let ˜r denote final wealth and W ~ the random return on the risky W asset, ρ is the nonrandom return on the riskless asset, W0 is initial nonrandom wealth, and α is the amount of initial wealth allocated to the risky asset. Pratt and Arrow show that Au(x) ≥ Av(x) for all x is equivalent to αu ≤ αv for all initial wealth levels and all risky assets. That is, A-P show that the more risk averse decision maker, when risk aversion is measured by Au(x), reacts to a risk in a portfolio decision setting by investing more of wealth in the riskless asset and less in the risky asset. This finding is given below as characterization (8). (8)   αu ≤ αv for all W0, ˜r and ρ. Similar results have been demonstrated in many other decision settings by researchers other than Arrow or Pratt. For example, Sandmo (1971) shows that for a competitive firm that chooses its output level when facing a random output price, more risk averse firms, those with a larger Au(x), choose a smaller output level. Similarly, when purchasing coinsurance, Smith (1968) and Mossin (1968) show that one consumer is more risk averse than another as measured by Au(x) if and only if that consumer chooses a higher 4

 ortfolio decisions involving a single risky asset can be dealt with in an EU decision model. It is the case of multiple P risky assets that M-V analysis is more tractable than is EU analysis.

The Theory of Risk and Risk Aversion

degree of coinsurance than does the less risk averse consumer. These and other such findings in a wide variety of decision settings lend considerable support for using Au(x), the A-P measure of absolute risk aversion, both as a measure of the intensity or strength of risk aversion, and as a way to compare risk aversion levels across decision makers. There are some technical matters concerning Au(x) that are of interest. These stem from analysis presented by Pratt (1964) showing how u(x) can be obtained from Au(x) ′′ u (x) using a three step procedure. In the first step, [−Au (x)] = ′ is integrated to obtain u (x) ln [u(x)]+c1, where c1 is the constant of integration. In step two, the result from step one is used as the exponent for base e.This yields u(x)·ec1. Finally in step three, the result from step two is integrated giving the final outcome of the three steps as u(x) · ec1 + c2. This three step procedure indicates how one can find the u(x) that leads to a particular Au(x). The process, however, is not as easy to carry out as it is to describe because the integration involved may not lead to a closed form solution. This three step process can be carried out in principle as long as one assumes that either  u (x) > 0 or u(x) < 0 for all x so that Au(x) never involves division by zero and is everywhere defined. Of course, in economic analysis u(x) > 0 is the common assumption. As a practical ′ c matter, however, the integration of [−Au(x)] in step one, or integrating [u (x)·e 1] in step three may not lead to a solution in closed form. As a consequence, some EU risk preferences can be represented by Au(x), but one cannot write down either the u(x) or u(x) associated with those risk preferences. It is also the case that there are EU risk preferences that can be represented by a marginal utility function u(x), and the corresponding u(x) is not of closed form. These things can be important to keep in mind when choosing functional forms to represent the risk preferences of decision makers in more applied settings. Many functional form choices exist for Au(x), fewer for u(x), and fewer forms yet for u(x). Saha (1993), Conniffe (2007a,b) and Meyer (2010) provide additional discussion of these issues. The next section discusses the two primary forms for u(x) considered in the literature.

3.4.4  Constant and Decreasing Absolute Risk Aversion The findings presented in Sections 3.4.1 and 3.4.2 support using the magnitude of the A-P measure of absolute risk aversion to measure the intensity of risk aversion and using Au(x) to compare the risk aversion levels of decision makers. There are a number of other topics concerning risk aversion that can be discussed. This section focuses on one of the more important ones involving decreasing absolute risk aversion (DARA). As part of this discussion, the two most commonly employed functional forms for u(x) are presented, and a general class of utility functions, the hyperbolic absolute risk averse (HARA) class, is defined. To determine functional forms for utility functions that display DARA, starting with those where Au(x) is a constant is useful.These constant absolute risk averse (CARA) utility functions take the form −cx

u(x) = −e

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where Au(x) = c is the measure of absolute risk aversion. Decision makers who are constant absolute risk averse have risk and probability premiums that do not depend on the initial wealth. At the very beginning of their discussion of measuring risk aversion both Arrow and Pratt conjecture that the slope of the absolute risk aversion measure is negative. The assumption that Au(x) ≤ 0 is referred to as decreasing absolute risk aversion with acronym DARA. This assumption can be defended in many ways, but most support for the assumption in the economics literature comes from showing that the assumption leads to sensible predictions in a variety of models. A sample of these comparative static predictions is given in Section 3.4.5. DARA is the assumption that, as the initial starting point for a decision maker becomes larger, the reaction to a risk of fixed size falls. For instance, if a decision maker begins with a larger amount of wealth, the risk premium or probability premium associated with a fixed gamble z˜ is smaller. Similarly, a decision maker with more wealth to invest will allocate more to the risky asset. Each of these statements assumes that the size of the risk itself is unchanged, and thus under DARA an increase in the starting position causes the decision maker to evaluate the alternatives using utility values from a less risk averse portion of the utility function. More formally, the following theorem is similar to one found in Pratt’s work.

Theorem 3.2:  The following statements are equivalent. (1) Au(x) is decreasing for all x. (2) π(x) is decreasing for all x and all gambles z˜. (3) φ(x, h) is decreasing in x for all x and all h. (4) α(W0) is increasing in W0 for all W0, ˜r and ρ. To determine which utility functions exhibit DARA, the following explicit writing ′ out of Au (x) is used: ′

Au (x) =

′′′

′′

−u (x) · u (x) + (u (x)) ′

(u (x))

2

2

.

This expression makes a number of things quite obvious. First, it is the case that u‴(x) ≥ 0 ′ is a necessary, but not sufficient condition for Au (x) to be negative. The assumption that u‴(x) ≥ 0 is referred to as prudence, thus it is the case that DARA implies prudence. Prudence is discussed again in Section 3.4.7. Also, since u‴(x) = 0 under quadratic utility, a quadratic utility function does not display DARA, but is increasing absolute risk averse instead. This property of a quadratic function is often mentioned as a negative feature and an argument against using quadratic functions as N-M utility functions. Quadratic utility functions also display the even worse feature that they must eventually have a negative slope.

The Theory of Risk and Risk Aversion

A form for Au(x) that exhibits DARA, and that is commonly used, is Au (x) = αx for α > 0. This family of risk aversion measures, indexed by α, has the feature that Au(x) is always decreasing, but does not become zero for any finite value for x. These Au(x) represent risk preferences for which the reaction to a given risk diminishes as the starting position is increased, but risk aversion never disappears.The form for the utility function leading to these absolute risk aversion measures is 1−α

x for all α � = 1 1−α u(x) = ln x for α = 1. u(x) =

These risk preferences and this particular functional form for utility also display constant relative risk aversion (CRRA) discussed next.

3.4.5  Relative Risk Aversion In addition to defining and supporting the use of the absolute risk aversion measure Au(x), Arrow and Pratt also define relative or proportional risk aversion. This relative risk aversion measure, Ru(x), differs from Au(x) in the way it measure the intensity of risk aversion, but Au(x) and Ru(x) do not yield different partial orders over decision makers in terms of their levels of risk aversion. That is, Ru(x) ≥ Rv(x) for all x > 0 is equivalent to Au(x) ≥ Av(x) for all x > 0. When measuring the intensity of risk aversion, Ru(x) uses the reaction to risk measured as a proportion of the initial wealth rather than as an absolute value. Ru(x) is defined by ′′

Ru (x) =

−x · u (x) ′

u (x)

= x · Au (x).

R(x) is supported as a measure of the intensity of risk aversion using arguments similar to those that are used for A(x). For instance, the proportional risk premium, π*(x), 2 defined to be π(x) x , can be related to the size of Ru(x) and σz using the same Taylor’s series approximation as was employed earlier for π(x). Doing this gives ∗

2

π (x) ≈ (1/2) · Ru (x) · σz .

Relative risk aversion is only sensibly defined when x > 0 is assumed. This implies that in portfolio decisions, for instance, the random outcome x is selected to be return5 5 The return is equal to one plus the rate of return. In the typical portfolio model, final wealth W

= α · ˜r + (W0 − α) · ρ

where W0 is initial wealth and α is the amount invested in the risky asset. In this formulation, both ˜r and ρ are returns to the risky and riskless assets respectively.These variables represent the gain or loss to the investment plus the return of the amount initially invested. If losses are limited to the amount invested, then the random return ˜r cannot be smaller than zero. Rate of return gives the change in the initial investment expressed as a percentage change and can be negative.

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rather than rate of return so that x > 0 is an acceptable restriction. Relative risk aversion is the elasticity of marginal utility, u(x), and thus is unit free. ′

Ru (x) =

−d(u (x))/u (x) . dx/x

Being unit free is a valuable property for a risk aversion measure when comparing findings concerning the magnitude of risk aversion across various studies where different units of measure for x are employed. A theorem similar to Theorem 3.1 follows immediately.

Theorem 3.3:  The following statements are equivalent. (1) u(x) is at least as risk averse as v(x). (2) Ru(x) ≥ Rv(x) for all x > 0. ′ (3) u(x)  = θ(v(x)) where θ (·) ≥ 0 and θ″(·) ≤ 0. ∗ ∗ (4) πu (x) ≥ πv (x) for all x and all gambles z˜. As noted earlier, Arrow and Pratt conjecture that Au(x) is likely to be a decreasing function. They also argue that Ru(x) is likely to be an increasing function. This assumption is referred to as increasing relative risk aversion (IRRA). Ru(x) is a constant, Ru(x) = α, for the family of utility functions where Au (x) = αx for α > 0 that was just discussed. Analysis since Pratt and Arrow lends considerable support for the assumption of DARA, especially when the outcome variable is wealth, but the findings are mixed concerning IRRA for wealth or for any other outcome variable. In fact, when the outcome variable is consumption rather than wealth, the opposite finding is often claimed for Ru(x); that is, Ru(x) is often estimated to display decreasing relative risk aversion (DRRA) when x represents consumption. A decreasing Ru(x) implies an even more steeply decreasing Au(x). Habit formation utility is a good example of such a DRRA utility function for consumption. Constantinides (1990), and Campbell and Cochrane (1999) use such habit formation utility functions. Meyer and Meyer (2005, 2006) provide a comprehensive review of this topic and literature. The two families of utility functions discussed so far are special cases of a more general family termed the hyperbolic absolute risk averse (HARA) family. Members of this general family of utility functions have the characteristic that risk tolerance, Tu (x) = Au1(x), is a linear function of x. A general form for a HARA form utility function is  (1−γ) (γ) x . u(x) = +η (1 − γ) γ From this utility function one can easily determine that Tu (x) = γx + η and γ . Letting γ go to infinity allows the CARA family of risk aversion Au (x) = x+γ·η

The Theory of Risk and Risk Aversion

measures with risk aversion measure 1/η to be obtained. The CRRA class of utility functions results from choosing η = 0. Gollier (2001) provides a more detailed discussion of special cases of the HARA family. A modified version of his notation was used here.

3.4.6  Comparative Statics Using Risk Aversion Measures ′

The derivatives of risk aversion, measured by either Au (x) or Ru (x), are important factors in determining how the choice made by a decision maker changes as the starting wealth changes. This is true in most decision models where expected utility is maximized. Two of the best-known results are presented next to illustrate the comparative statics of risk aversion, and how these comparative static findings are demonstrated.The two examples also illustrate the difference between and the absolute and relative risk aversion intensity measures. First consider a perfectly competitive firm choosing output level x to maximize expected utility from profit given by π ˜ = p˜ · x − c(x) − B. In this model, which follows Sandmo’s (1970) notation, output price p˜ is random with CDF F(p) with support in [a, b], c(x) is the total variable cost function for the firm and B is total fixed cost. The firm chooses output level x to maximize

b

u(π)dF(p).

a

The first order condition for this maximization is:

b

u (π)(p − c (x))dF(p) = 0.

a

The second order condition is satisfied when the firm is risk averse and variable costs are convex. Now consider what happens when fixed cost B changes. This change is a downward shifting or reduction of the starting point for the decision maker, equivalent to a reduction in initial wealth. Differentiating the first order expression with respect to B gives  b ′′ ′ −u (π)(p − c (x))dF(p). a

This expression can be rewritten so that the absolute risk aversion measure and the first order expression each are present in the expression

a

b

Au (π)[u (π)(p − c (x))]dF(p).

Integrating by parts one time gives

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a

s

−Au (π) ·

s

u (π)(p − c (x))dF(p)ds.

a

Using the first order condition, it is the case that as u′ (π)(p − c′ (x))dF(p) ≤ 0 for all ′ s, and therefore if Au (π) is assumed to be negative (positive), this expression is negative (positive). This shows that when B is increased, a firm always decreases its choice ′ of output level if Au (π) ≤ 0. The assumption of DARA implies that increased fixed costs cause firms to reduce output. The logic is quite simple. DARA combined with an increase in B implies that the firm is evaluating its output decision assuming lower profit levels and hence using a more risk averse portion of its N-M utility function u(π). Since the conclusion reverses when Au(π) is an increasing function instead, this result is considered to be support for the DARA assumption. Much of the comparative static analysis in EU decision models follows this general pattern and involves proofs using a similar methodology. A second example illustrates the comparative static effect of Ru(x) in the important and often examined single risky and riskless asset portfolio problem. Let the final outcome variable be wealth given by  = W0 (α · ˜r + (1 − α)ρ) W

where W0 is nonrandom initial wealth, and ˜r is the random return from investing in the risky asset. The CDF for ˜r is denoted F(r) whose support is in some interval [0, b] and ρ is the nonrandom return from the riskless asset. In this version of the portfolio decision, α is the proportion of wealth invested in the risky asset. ˜r > 0 and ρ > 0 and α in  > 0. This model differs from that used in Section 3.4.2 [0, 1] are assumed so that W when discussing characterization (8) of more risk averse in that α is now a proportion of initial wealth rather than an absolute amount. This change implies that W0 and the random parameter ˜r are multiplied by one another rather than added together. The investor is assumed to maximize Eu(W) which is given below.  b u(W)dF(r) 0

The first order condition for this maximization is  b ′ u (W)W0 (r − ρ)dF(r) = 0. 0

The second order condition is satisfied if the investor is risk averse. Now the shifting of the starting point is increasing the initial wealth W0. Differentiating the first order expression with respect to W0 yields

The Theory of Risk and Risk Aversion

b

′′

u (W)W(r − ρ)dF(r)

0

which can be rewritten as

0

b

−Ru (W)u (W)(r − ρ)dF(r).

As in the previous example, integrating this expression by parts can be used to show that if R(W) is increasing (constant, decreasing), the proportion of initial wealth allocated to the risky asset decreases (stays the same, increases) as initial wealth increases. These two detailed examples are each special cases of a general decision model where the decision maker chooses a variable α to maximize expected utility from z(α, x˜ , β). In this notation x˜ is a random parameter with CDF F(x) with support in the interval [a, b], and β represents nonrandom parameters, possibly more than one, and α represents a decision variable. A variety of questions can be posed within this framework. One can ask how α changes as a parameter β changes, including the starting wealth the parameter just discussed in the two detailed examples. Other parameters, including the random parameter x˜ can also be changed and the analysis follows a similar methodology. Kraus (1979) and Katz (1981) ask how does α change as x˜ becomes riskier in a specific way. The general model they pose omits the notation for the nonrandom β since these parameters are held constant in their analysis. The literature contains far more comparative static analyses of both general and specific decisions than can be discussed further here. Other measures of risk aversion that preserve the partial order over decision makers have been suggested. These are often defined because an assumption concerning the new risk aversion measure is particularly well suited for a particular comparative statics question. One such measure is proposed by Menezes and Hanson (1970). Using their notation, the measure of partial relative risk aversion is given by ′′

Pu (t; w) =

−t · u (t + w) ′

u (t + w)

= t · Au (t + w).

This measure is a partial measure of relative risk aversion in that the variable multiplying Au(t + w) is not the whole argument of Au(·), but only a portion of it. This measure proves to be particularly useful for comparative static analysis when the initial wealth is kept fixed, but the scale of the risk is altered. Like Ru(x), this partial relative risk aversion measure does not define a different ordering over decision makers in terms of risk aversion. The main reason for defining either relative risk aversion or partial relative

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risk aversion is to allow a convenient interpretation of a particular assumption concerning risk preferences. It is easier to interpret the statement that R(x) is a constant than it is to interpret the statement that A(x) takes the form A(x) = αx even though these are equivalent statements.

3.4.7  Ross’s Strongly More Risk Averse Order For a number of important decisions, it is the case that the assumption that Au(x) ≥ Av(x) for all x is not sufficient to predict how one decision maker chooses differently than another. This is especially true in decision models where there is more than one source of randomness. Ross (1981) proposes a different definition of increased risk aversion, shows that his definition implies that of Arrow and Pratt, and uses the term strongly more risk averse to describe the order over decision makers his definition yields. Ross then shows that when one decision maker is strongly more risk averse than another, comparative static predictions that are ambiguous under the assumption that Au(x) ≥ Av(x), for all x, become determinate. Formally, Ross provides the following definition of when a decision maker with utility function u(x) is strongly more risk averse than another with utility function v(x).

Definition 3.2:  u(x) is strongly more risk averse than v(x) if there exists a λ > 0 such that ′

′′

u (x) ′′

v (x)

≥λ≥

u (y) ′

v (y)

for all x and y.

To understand where the Ross order is most applicable and the additional requirements this stronger order imposes, recall that Arrow and Pratt define the risk premium for a decision maker starting with nonrandom wealth x and associated with gamble z˜ whose mean value is zero, using the equation u(x − π) = Eu(x + z˜ ).

For this π(x), Theorem 3.1 indicates that Au(x) ≥ Av(x) for all x is necessary and sufficient for πu(x) ≥ πv(x) for all x and all gambles z˜. What is important to note is that in this definition of π(x), the starting point or initial wealth x is not a random variable. Ross considers a very similar risk premium, a fixed amount to be subtracted from the decision maker’s initial wealth, but allows the initial wealth to be a random variable x˜ rather than requiring it to be nonrandom. Since both z˜ and x˜ are random, Ross assumes that E[˜z|x] = 0 for all x. This replaces the assumption that E[˜z] = 0 made by Arrow and Pratt. Formally, the Ross risk premium is the amount the mean value of z˜ is discounted in order to sell it when the decision maker begins with random wealth x˜ . This is the same terminology as used to describe the Arrow and Pratt risk premium π(x). The difference is that the initial wealth is no longer assumed to be certain. Formally, π is defined by the equation

The Theory of Risk and Risk Aversion

Eu(˜x − π) = Eu(˜x + z˜ ).

Ross shows that a stronger definition of more risk averse is required to determine when πu ≥ πv for all random initial wealth positions, x˜ , and all gambles z˜ satisfying E[˜z|x] = 0. He shows that the following three statements are equivalent ways to define when a decision maker with utility function u(x) is strongly more risk averse than another with utility function v(x).

Theorem 3.4:  The following three conditions are equivalent. (1) There exists a λ > 0 such that

′′

u (x) ′′ v (x)

≥λ≥

u (y) ′ v (y)

for all x and y. ′

(2) There exists a λ > 0 and a function φ(x) with φ (x) ≤ 0 and φ″(x) ≤ 0 such that u(x) = λ·v(x) + φ(x) for all x. (3) For all random x˜ and z˜ such that E(˜z|x) = 0, Eu(˜x + z˜ ) = Eu(˜x − πu ) and Ev(˜x + z˜ ) = Ev(˜x − πv ) imply that πu ≥ πv. The Ross definition of strongly more risk averse differs from other alternatives to absolute risk aversion such as relative or proportional risk aversion in that it provides a different and stronger partial order over decision makers in terms of their risk aversion levels. With the stronger risk aversion order, Ross is able to extend many theorems to more general settings. For instance, consider a portfolio decision with two assets where both assets are risky and yield a random return. If the return on the first asset is denoted x˜ and the return on the second y˜ , then the portfolio allocation decision is to choose α to maximize expected utility from  = W0 (α · y˜ + (1 − α)˜x) = W0 (˜x + α(˜y − x˜ )). W

Letting z˜ = (˜y − x˜ ) and assuming that E(˜z|x) ≥ 0 for all x so that the second asset has return y˜ which is both larger and riskier than z˜ Ross shows that when u(W) is strongly more risk averse than v(W), then αu ≤ αv for all such y˜ and x˜ . That is, the strongly more risk averse decision maker chooses to include more of the less risky asset and less of the more risky asset in the two asset portfolio. Many other models can be generalized to include multiple random variables and a random starting point and yet comparative static findings can be obtained if the strongly more risk averse assumption is made. The Ross fixed risk premium was generalized to a random risk premium by Machina and Neilson (1987) who use this to provide an additional characterization of Ross’s strongly more risk averse order. Pratt (1990) points out several negative and overly strong aspects of the Ross order. In addition, because the strongly more risk averse order allows a random starting point, higher order measures of risk aversion use the Ross methodology in their formulation. Modica and Scarsini (2005), Jindapon and Neilson (2007) and Denuit and Eeckhoudt (2010a,b) use this feature to discuss higher order measures of risk aversion.

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3.4.8  Other Risk Aversion Terms and Measures Rather than additional conditions on the first and second derivatives of u(x), other contributions to the discussion of risk aversion add conditions that restrict the sign of higher order derivatives of u(x). These conditions provide another way to strengthen the requirement that Au(x) ≥ Av(x) for all x so that additional comparative static findings can be obtained. Ekern (1980) does this in a very general fashion defining nth degree risk aversion as follows.

Definition 3.3:  Decision maker with utility function u(x) is nth degree risk averse if

(−1)n−1 · u(n) (x) > 0 for all x. In this notation, u(n) denotes the nth derivative of u(x). Ekern uses this definition and a related one concerning nth degree risk increases to show that G(x) has more nth degree risk than F(x) if and only if every nth degree risk averse decision maker prefers F(x) to G(x). This is discussed more completely in Section 3.5.4, where the measurement of higher order risk and risk increases is discussed, and higher order stochastic dominance is defined. At least two of the higher order derivatives of u(x) have been used to define additional terms that describe risk preferences of decision makers. Kimball (1990) focuses on the sign of u‴(x). He notes that when u‴(x) ≥ 0, marginal utility is convex and many of the Arrow and Pratt derivations for u(x) can be reapplied and interpreted for ′′′ u(x) instead. The sign of u‴(x) and P(x) = −u′′ (x) are used to define and measure the u (x) intensity of prudence.

Definition 3.4:  A decision maker with utility function u(x) is prudent if u‴(x) ≥ 0

for all x. As mentioned when discussing DARA, being prudent is a necessary but not sufficient condition for a decision maker to be decreasingly absolute risk averse. Kimball shows that this weaker restriction, prudence, is sufficient to generate interesting comparative static predictions in several decision settings. The initial discussion of prudence focuses on the effect of the introduction of randomness in the amount available to spend in the second period of a two period consumption model. The question is whether this future randomness increases or decreases saving from the first period to the second. Kimball shows that prudent decision makers, those with u‴(x) ≥ 0 for all x, save more as a result of the introduction of risk into the amount available in the second period. In another paper, Kimball (1993) also defined temperance using the sign of the fourth order derivative of u(x). He also defines standard risk aversion as the property that both Au(x) and Pu(x) are decreasing functions. There is still considerable discussion concerning the correct way to measure the intensity of prudence or temperance. Eeckhoudt and Schlesinger (2006) use the combining or separating of two simple “bads,” a reduction

The Theory of Risk and Risk Aversion

in initial wealth or the addition of a zero mean random variable, to further characterize prudence and temperance and other higher order derivatives of u(x). Others using the signs of higher order derivatives of u(x) include Menezes et al. (1980) who use the sign of the third derivative when defining downside risk aversion. Those adding to this discussion include Crainich and Eeckhoudt (2008), Keenan and Snow (2002, 2009) and Liu and Meyer (2012). As was mentioned at the outset, the A-P risk aversion measure Au(x) assumes that the N-M utility function u(x) has a single argument rather than a vector of arguments. At the same time, there is nothing in the axioms of expected utility that disallow outcome variables that are vectors. The discussion here of measuring risk aversion concludes with a brief review of the findings concerning extending measures of risk aversion to vector valued outcomes. An initial discussion of this topic occurs in the two period consumption model where utility is assumed to take the form u(c1, c2). For this decision model, the assumption is that first period consumption is a choice variable and is nonrandom, with second period consumption resulting from the return on the saving from the first period plus additional resources made available in the second period. It is typical to assume that either the return on saving or the additional resources that arrive in the second period is a random variable. Thus, in this model, c1 is chosen first and is not random, and c2 is random because its value is partially determined by random return on saving or random endowment in the second period. Typical notation is for W1 and W2 to represent the endowment in periods one and two respectively. The amount consumed in period one, c1, is chosen and the residual, (W1 − c1) is saved for second period consumption. Second period consumption is given by c2 = W2 + (W1 −c1) · r. To introduce randomness, either W2 or r, but not both, is assumed to be random. In this two period consumption model, since only c2 is random, the 22 ratio −u u2 can be used as a measure of absolute risk aversion. This measure has many of the same general properties as the A-P measure Au(x). Sandmo (1970) and Leland (1968) use such a measure and present results in the two period consumption model. For the more general case, where utility is a function of a vector of outcome variables, Kihlstrom and Mirman (1974) present a thorough discussion and draw a quite negative conclusion. Using an example, they show that it is not possible to disentangle preferences for bundles of goods in settings without randomness from preferences for risk. Thus, they settle on a general definition of more risk averse that is similar to characterization (3) from Theorem 3.1 and listed below for convenience. ′

′′

(3) u(x) = θ(v(x)) where θ (·) ≥ 0 and θ (·) ≤ 0. Kihlstrom and Mirman propose exactly this same condition as the definition of u(x) being more risk averse than v(x) with the interpretation that x can be a vector of dimension higher than one.

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function v(x) if there exist a θ(·) such that θ(·) ≥ 0 and θ″(·) ≤ 0 and u(�x) = θ(v(�x)). To compare the risk aversion of decision makers using this definition requires that the decision makers display exactly the same ordinal preferences in settings without randomness. This severely limits the applicability of the definition yet a replacement has not been proposed.

Definition 3.5:  For x in Rn, utility function u(x) is at least as risk averse as utility

3.5 INCREASES IN RISK AND STOCHASTIC DOMINANCE 3.5.1 Introduction Soon after Arrow and Pratt defined and supported Au(x) as a useful measure of risk aversion, the attention of researchers shifted to the other half of the expected utility calculation, how to categorize and describe the random variable or CDF representing the likelihood of the various possible outcomes that could occur. Two different but related questions are discussed. The more general question asks what are the necessary and sufficient conditions for one random variable or CDF to be preferred or indifferent to another by all decision makers with risk preferences restricted to some particular set? That is, what are the necessary and sufficient conditions for F(x) to be preferred or indifferent to G(x) for all EU decision makers with N-M utility function in some set ? First degree, second degree and other forms of stochastic dominance are discussed using this general framework. Hadar and Russell (1969) and Hanoch and Levy (1969) are early contributors in this area. The second question is more narrowly focused and is directly related to the question of defining or measuring risk aversion. This question asks when is one random variable or CDF riskier than another? Rothschild and Stiglitz (1970) provide an answer in the form of a definition of increasing risk. The discussion begins with first degree stochastic dominance.

3.5.2  First Degree Stochastic Dominance Hadar and Russell (1969) and Hanoch and Levy (1969) each provide what are now known as first and second degree stochastic dominance conditions, and use these conditions to define one random variable stochastically dominating another in the first and second degree. These conditions use CDFs to represent random variables and place restrictions on [G(x) − F(x)]. These conditions could also be written as restrictions on the random variables themselves. Because of its simple nature first degree stochastic dominance is reviewed first and the discussion of second degree dominance is deferred to Section 3.5.3 following the presentation of the definition of increasing risk.

Definition 3.6:  F(x) dominates G(x) in the first degree if G(x) ≥ F(x) for all x. This definition is formulated without making the assumption that the decision maker maximizes EU.That is, first degree stochastic dominance (FSD) is defined outside

The Theory of Risk and Risk Aversion

of the expected utility decision model context. The implications in an EU decision model, however, provide an interpretation for FSD and support for the definition.

Theorem 3.5:  F(x) dominates G(x) in the first degree if and only if F(x) is preferred or indifferent to G(x) by all EU decision makers with N-M utility functions satisfying u(x) ≥ 0 for all x.

First degree stochastic dominance can be thought as the extension of the concept of “larger” as it applies to real numbers. FSD is “larger” for random variables whose outcomes are real numbers. Random variables can become larger in at least two distinct ways. Without changing the likelihoods, the outcomes themselves can be increased, or alternatively, without changing the outcomes, the likelihood of the larger outcomes can be increased and the likelihood for the smaller outcomes decreased. Each of these changes increases the size of a random variable and is represented by a first degree stochastic dominant increase as given in Definition 3.6. Theorem 3.5 indicates that all decision makers who prefer larger outcomes, that is, those with u(x) ≥ 0, agree unanimously that this restriction, G(x) ≥ F(x) for all x, captures their preference for larger outcomes and larger random variables. The proof of Theorem 3.5 is quite simple and involves integration by parts and steps similar to the comparative static analysis discussed in Section 3.4.5. FSD yields a partial order over CDFs. It only provides a ranking for random variables whose CDFs do not cross.

3.5.3  Increases in Risk Rothschild and Stiglitz (1970) (R-S) present a definition of what it means for one random variable to be riskier than another. In fact, they present three distinct definitions and then show that the three seemingly different definitions of one random variable being riskier than another are equivalent. As is the case for the Arrow and Pratt measure of risk aversion, the result of this analysis is a single definition of increasing risk with multiple ways to characterize the definition. The R-S definition only applies to random variables whose mean values are the same.The R-S definition of increasing risk is different from variance, but does imply that the riskier random variable has higher variance. The first definition of one random variable being riskier than another that R-S propose is one that involves the addition of “noise” to an existing random variable, and is similar to definitions used in the literature concerned with the analysis of signals. Random variable y˜ is defined to be riskier than x˜ when y˜ is obtained from x˜ by adding noise.

Definition 3.7A:  y˜ is riskier than ~x if y˜ is equal in distribution to x˜ + z˜, where

E(˜z|x) = 0. It is important to notice that the added noise, z˜, is uncorrelated with x˜ and since it has zero mean for each x, the mean of z˜ is zero. This implies that y˜ and x˜ have the same mean. It need not be the case that z˜ is independent of x˜ .

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The second definition of y˜ riskier than x˜ is based on the most basic concept of risk aversion in EU decision models, namely the concavity of u(x). Random variable y˜ is said to be riskier than x˜ if x˜ is preferred or indifferent to y˜ by all decision makers with concave utility functions, those who are risk averse. No assumption is made concerning the sign of u(x). In the formal statement of this definition, random variables y˜ and x˜ are represented by CDFs G(x) and F(x), respectively. This second definition argues that one can define an increase in risk as a change in a random variable that reduces expected utility for all risk averse persons, those who avoid risk.

Definition 3.7B:  y˜ is riskier than x˜ if

b

b

a a u(x)dG(x) for all concave u(x). In stating this definition, the assumption is made that the supports of the random variables lie in a bounded interval. Rothschild and Stiglitz assume that this bounded interval is [0, 1]. The third R-S definition of increasing risk introduces and defines a term that is important in its own right. This term has become part of an economist’s vocabulary. This third definition indicates that y˜ is riskier than x˜ if the probability or density function for y˜ has “more weight in the tails” than that for x˜ . To make “more weight in the tails” precise, R-S define a mean preserving spread, and this term is now in common usage. A mean preserving spread (MPS) is a taking of probability mass from its current location in the distribution of probability, and redistributing that mass to locations to the right and to the left of the original location. This redistribution is carried out in such a way as to preserve the mean of the random variable. Probability mass is taken from center of the initial distribution, where “center” can be anywhere that probability mass is currently distributed, and this mass is redistributed to both lower and higher outcomes, keeping the mean outcome fixed. When such a MPS is carried out, R-S argue that it is natural to think of the new random variable as riskier than the original. A formal definition of a mean preserving spread for a discrete random variable is provided by Rothschild and Stiglitz. A definition for continuously distributed random variables is given here. The notation follows that of Diamond and Stiglitz (1974) and Meyer (1975). Assume that random variable x˜ has probability density function f(x). Define as a mean preserving spread the function s(x). s y ˜ x ˜ Definition 3.7C:   is riskier than if a [G(x) − F(x)]dx ≥ 0 for all s in [a, b] and b . [G(x) − F(x)]dx = 0 a

u(x)dF(x) ≥

The conditions on [G(x) − F(x)] specified in this third definition provide useful test conditions that can be readily applied to any pair of CDFs and have become the most common way to specify when one random variable is riskier than another.

The Theory of Risk and Risk Aversion

R-S show that these three quite different definitions are actually one and demonstrate the following theorem.

Definition 3.8  s(x) is a mean preserving spread if    i) ii)

b a

b a

s(x)dx = 0. x − s(x)dx = 0.

iii) s(x) changes sign twice from positive to negative back to positive.   iv) f(x) + s(x) ≥ 0 for all x. The first and fourth conditions together ensure that when s(x) is added to f(x), the result is a density function g(x). Condition ii) makes the change mean preserving. Condition iii) indicates that when s(x) is added to f(x), probability mass is shifted from its current location, where s(x) is negative, and some of this mass is moved to the left and some to the right. This occurs when s(x) is positive. A typical mean preserving spread is given in Figure 3.3. A single mean preserving spread leads to CDFs that cross exactly one time and the riskier CDF is positive first. The mean preserving condition implies that the area above the horizontal axis is the same size as that below the axis. This is illustrated in Figure 3.4. Assuming that the definition of increased risk is transitive and therefore a sequence of mean preserving spreads also leads to a riskier random variable, R-S determine the

s(x)

a

b

Figure 3.3  A typical mean preserving spread for density functions.

G(x) ─ F(x) a

b

Figure 3.4  A typical mean preserving spread for cumulative distribution functions.

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conditions on CDFs F(x) and G(x) for G(x) to be obtainable from F(x) by a sequence of MPSs. These conditions involve partial integrals of [G(x) − F(x)] and are the formal statement of the third definition of an increase in risk. Theorem 3.6:  The following statements are equivalent. (1) y˜ is equal in distribution to x˜ + z˜, where E(˜z|x) = 0.   (2) ab u(x)dF(x) ≥ ab u(x)dG(x) for every concave u(x).   (3) as [G(x) − F(x)]dx ≥ 0 for all s in [a, b] and ab [G(x) − F(x)]dx = 0.

The R-S definition of increasing risk has been accepted and used for more than forty years. No serious alternative has arisen in that time to replace it. This definition leads to a partial rather than complete order over random variables in terms of riskiness. The order is partial in two different senses. First, only the riskiness of random variables with the same mean can be compared. Second, even for random variables with the same mean, it is very possible that neither is riskier than the other. Attempts have been made to extend the definition to random variables whose means are different. Diamond and Stiglitz do this defining a mean utility preserving spread, replacing condition ii) in Definition 3.8 with  b ∗ ii) u(x) · s(x)dx = 0 a

which is the mean utility preserving condition. Meyer (1975) shows that this yields an order that is complete for all random variables whose cumulative distribution functions cross a finite number of times, but that this complete order is equivalent to the max–min order which ranks random variables according to the size of the worst possible outcome. The R-S definition of increasing risk is an ordinal concept. None of the three characterizations is used to measure the size of the risk associated with a random variable. This is in contrast to variance, which orders random variables completely in terms of riskiness, and also gives a numerical measure of the size of the risk. Aumann and Serrano (2008) use the reciprocal of the absolute risk aversion measure of the CARA decision maker who is just indifferent between taking and rejecting a gamble as a measure of the size of the riskiness of that gamble. They specifically examine gambles with a positive mean value and a positive probability that negative outcomes occur. For these gambles, they show that their measure of riskiness has several desirable properties, and compare it with other attempts to determine an index of riskiness.The Aumann and Serrano index provides a measure of riskiness that is an alternative to variance as a measure of riskiness. Finally, as with the measure of absolute risk aversion, the definition of increased riskiness is for random variables whose outcomes are single dimension rather than vectors.

The Theory of Risk and Risk Aversion

There is an extensive literature that uses the R-S definition of increasing risk (Rothschild and Stiglitz, 1971). Comparative statics analysis in a variety of decision models has examined the effects of R-S increases in risk on choices made, and on the welfare of the decision maker. It is often the case that analysis of these comparative static questions does not yield a determinate conclusion that applies to all R-S increases in risk, and therefore further restrictions on the change in riskiness have been proposed. Sandmo (1971) discusses risk increases that change x˜ by defining y˜ = α + θ · x˜ where α and θ are chosen so that the mean is not changed. θ > 1 implies that risk has increased. Meyer and Ormiston (1985) generalize this, defining strong increases in risk. Others, including Black and Bulkley (1989) and Kihlstrom et al. (1981), add to this literature. The literature was summarized and a very general result presented by Gollier (1995). A review of these refinements is beyond the scope of this chapter. A summary can be found in Gollier (1995) and Gollier and Pratt (1996).

3.5.4  Second Degree Stochastic Dominance Definition 3.7B is in the form of a stochastic dominance finding. That is, that definition provides an answer to a question of the form: what are necessary and sufficient conditions for one random variable to be preferred or indifferent to another by all decision makers in a specified group of decision makers? Rothschild and Stiglitz answer this question for the group of decision makers that include all those who are risk averse. Second degree stochastic dominance adds the restriction that the utility function must also satisfy u(x) ≥ 0 for all x; that is, second degree stochastic dominance considers all decision makers whose utility functions satisfy both u(x) ≥ 0 and u″(x) ≤ 0 for all x. The conditions for unanimous preference of one random variable over another for this group of decision makers define the well-known second degree stochastic dominance condition. Hadar and Russell (1969) and Hanoch and Levy (1969) each provide the definition. Like first degree stochastic dominance and the second R-S definition, these conditions use CDFs to represent random variables and place restrictions on [G(x) − F(x)]. Second degree stochastic dominance (SSD) is related to the R-S definition of an increase in risk. SSD differs from the definition of increasing risk because in addition to concavity of u(x), u(x) ≥ 0 is also assumed. With this smaller set of decision makers, the equal means condition of the R-S  definition of an increase in risk is no longer necessary, and therefore the condition that ab [G(x) − F(x)]dx = 0 is not part of the SSD definition. s Definition 3.9:  F(x) dominates G(x) in the second degree if a [G(x) − F(x)]dx ≥ 0 for all s in [a, b]. As with FSD, the implications in an EU decision model provide support for the definition.

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Theorem 3.7:  F(x) dominates G(x) in the second degree if and only if F(x) is preferred or

indifferent to G(x) by all EU decision makers with N-M utility functions satisfying u(x) ≥ 0 and u″(x) ≤ 0 for all x. One random variable can dominate another in the second degree because it is larger, less risky, or as a result of a combination of these two reasons. It is the case that when F(x) dominates G(x) in SSD that μF ≥ μG.

3.5.5  Other Forms of Stochastic Dominance A number of other forms of stochastic dominance have also been defined. Some have further restricted the set of decision makers under consideration by imposing sign conditions on higher order derivatives of u(x). Along with nth degree risk aversion discussed briefly in Section 3.4.7, Ekern also provides a definition of an nth degree risk increase and demonstrates a general stochastic dominance theorem. Definition 3.3 from Section 3.4.7 is repeated for convenience.

Definition 3.3:  Decision maker with utility function u(x) is nth degree risk averse if (−1)n−1·u(n)(x)  > 0 for all x.

Definition 3.10:  G(x) has more nth degree risk than F(x) if G(n)(x) ≥ F(n)(x) for all x in [a, b] and G(k)(b) = F(k)(b). These statements hold for k = 1, 2, …, n.

F(k)(x) denotes the kth cumulative probability distribution; that is, let F(1)(x) denote F(x), and higher order cumulative functions are defined by F(k) (x) = ax F(k−1) (s)ds, k = 2, 3,…. The relationship between Definitions 3.3 and 3.10 is also provided by Ekern who demonstrates the following theorem.

Theorem 3.8:  G(x) has more nth degree risk than F(x) if and only if every nth degree risk averse decision maker prefers F(x) to G(x).

This theorem contains the FSD result,Theorem 3.5, and a portion of the R-S increasing risk definition, Theorem 3.6, as special cases. Each of those findings restricts the sign of only one derivative of u(x) as does Definition 3.3. If one restricts the sign of each of the first n derivatives of u(x) rather than just the nth, then the following theorem results. Theorem 3.9 contains both Theorems 3.5 and 3.7 as special cases. This theorem also contains the third degree stochastic dominance result that was presented by Whitmore (1970).

Theorem 3.9:  F(x) stochastically dominates G(x) in the nth degree; that is G(n)(x) ≥ F(n)(x) for all x in [a, b] and G(k)(b) ≥ F(k)(b) for k = 1, 2, …, n if and only if every decision maker who is kth degree risk averse for k = 1, 2, …, n prefers F(x) to G(x).

The Theory of Risk and Risk Aversion

Other forms of stochastic dominance have imposed restriction on Au(x). DARA stochastic dominance, for instance, considers decision makers for whom u(x) ≥ 0, u″(x) ≤ 0 and Au(x) ≤ 0. The necessary and sufficient conditions for DARA stochastic dominance have not been characterized in a simple way. It is known, however, that DARA and third degree stochastic dominance are equivalent when the means of F(x) and G(x) are equal to one another. Fishburn and V   ickson (1978) and Liu and Meyer (2012) demonstrate this. Diamond and Stiglitz (1974) when defining mean utility preserving spreads and Meyer (1977) when defining stochastic dominance with respect to a function consider sets of decision makers defined by a lower bound on Au(x) that need not be zero; that is, all decision makers who are more risk averse than a reference decision maker where the reference decision maker need not be the risk neutral one. The following theorem provides a stochastic dominance condition for such sets of decision makers.

Theorem 3.10:  F(x) is preferred or indifferent to G(x) by all EU decision makers

N-M utility functions satisfying u(x) ≥ 0 and Au(x) ≥ Av(x) if and only if with s ′ a [G(x) − F(x)]v (x)dx ≥ 0 for all s in [a, b].

Fishburn (1974) discusses extending the concept of stochastic dominance to more than a pair-wise comparison. His work focuses on the case of two sets of random variables with CDFs {F1(x), F2(x), … Fn(x)} and {G1(x), G2(x), … Gn(x)}, respectively. For FSD the following theorem applies. A similar result holds for SSD.

Theorem 3.11:  For each u(x) with u(x) ≥ 0, there exists an i, i = 1, 2, … n, such that Fi(x) is preferred or indifferent to Gi(x) if and only if   ni=1 λi Fi (x) dominates  FSD for some (λ1, λ2, … λn) such that λi ≥ 0 and ni=1 λi = 1.

n

i=1 λi Gi (x)

in

3.6 MEAN VARIANCE ANALYSIS AND EXPECTED UTILITY Now that the EU decision model has been reviewed, the discussion returns briefly to the M-V decision model and the important question of when are these two decision models equivalent to one another in the way that they rank the random variables in the choice set. That is, when can an EU ranking of alternatives F(x) and G(x) be represented equivalently with a ranking function V(σ, μ)? Many have addressed this question, especially during the time period when the two decision models were each considered to be an interesting alternative in theoretical economic modeling of risky decisions. A very good summary of this literature is given by Baron (1977) and a portion of the material here comes from that paper. The main and simple answer to this equivalence question is that if there are no restrictions placed on the set of random variables to be ranked, it is necessary and sufficient that the N-M utility function u(x) be a quadratic function. Baron attributes this result to Markowitz (1959). If one adds the Borch requirement, discussed in section 3.2.2, that the utility function represent preferences where more is preferred to less, then

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even with quadratic utility functions the random variables that can be ranked have supports that are severely restricted. To see this assume that u(x) takes the form u(x) = x + c ·x2 for c < 0. For this quadratic utility function, the maximum outcome possible must 1 be less than −2c for u(x) to be nonnegative. Therefore if risk aversion is allowed, so that c > 0, the random variables being ranked must have supports that lie in intervals trun1 cated on the right by −2c . This makes it difficult to model wide ranges of risk aversion. All other answers to the equivalence question restrict the set of random variables to be ranked at the outset. When the choice set is restricted to normally distributed random variables, Chipman (1973) provides a very complete and detailed discussion. With the normality assumption, the utility function must be bounded for expected utility to be finite. Also, the M-V utility function over variance and mean T(σ2, μ) satisfies a particular differential equation given by 2

∂T ∂σ

2

2

=

∂ T ∂µ

2

.

Also for normally distributed random variables, Freund (1956) shows that the EU ranking function U(F) can be reduced to a simple V(σ, μ) whenever the N-M utility functions u(x) are of the CARA form. That is, Freund shows if u(x) = −e−cx and x˜ is normally distributed, the EU ranking function U(F) can be reduced to the M-V utility function V(σ, μ)=μ − λ·σ2, a commonly used and simple functional form for V(σ, μ). More recently Sinn (1983) and Meyer (1987) show that when the set of random variables to be ranked is restricted to include only alternatives which are location and scale transformations of one another, that is elements from the same location and scale family, then an EU ranking of these alternatives can be represented by V(σ, μ).

Definition 3.11:  Random variables {x1, x2, xn, ….} are in the same location and

scale family if for each xi and xj it is the case that xi = d a + b ·xj where b > 0. It is typical to choose to focus on a particular element x* from the location and scale family whose mean equals zero and variance equals one. For this x*, each xi = dμi+σi· x*. Here, μi and σi are the mean and standard deviation of xi. The set of all normally distributed random variables is a location and scale family as is the set of all uniformly distributed random variables. There is an infinity of such families. Location and scale families of random variables can be generated by picking any random variable with mean zero and variance equal to one as x* and then forming the remaining elements of the family with mean μi and standard deviation σi using the equation xi = dμi + σi· x* to generate those random variables. Obviously only a few such families have been named. When the set of random variables to be ranked are from a location and scale family, the M-V ranking function V(σ, μ) and the slope of indifference curves in (σ, μ) space, S(σ, μ), have several interesting properties.Among the more important of these properties are:

The Theory of Risk and Risk Aversion

Property 1:  V(σ, μ) is concave if u(x) is concave. Property 2:  Su(σ, μ) ≥ Sv(σ, μ) for all (σ, μ) if u(x) is more risk averse than v(x). ∂S Property 3:  ∂µ ≤ 0 if u(x) is decreasingly absolute risk averse. One reason why focusing on random variables from a location and scale family is a worthwhile exercise is that for decision models with only one random variable, and whose outcome variable is linear in that random variable, all possible random outcome variables automatically form a location and scale family. For instance in the single risky asset portfo~  = W0 (α · ˜r + (1 − α)ρ), all possible W lio choice model where the outcome variable W that can be formed by choosing different values for α, or when the parameters ρ or W0 are varied, are a location and scale family of random alternatives. Thus, this model and many others like it can be analyzed using either the M-V or EU decision model framework.

ACKNOWLEDGMENT Helpful comments from the editors and Liqun Liu are gratefully acknowledged.

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